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In this paper we compute the homotopy groups of the symplectomorphism groups of the three-, four- and five-point blow-ups of the projective plane (considered as monotone symplectic Del Pezzo surfaces). Along the way, we need to compute the homotopy groups of the compactly supported symplectomorphism groups of the cotangent bundle of $RP^2$ and of $C^∗ ×C$. We also make progress in the case of the $A_n$-Milnor fibres: here we can show that the (compactly supported) Hamiltonian group is contractible and that the symplectic mapping class group embeds in the braid group on n-strands.
We show that the Calabi homomorphism extends to some groups of homeomorphisms on exact symplectic manifolds. The proof is based on the uniqueness of the generating Hamiltonian (proved by Viterbo) of continuous Hamiltonian isotopies (introduced by Oh and Muller).